Superluminality and UV completion

Abstract

The idea that the existence of a consistent UV completion satisfying the fundamental axioms of local quantum field theory or string theory may impose positivity constraints on the couplings of the leading irrelevant operators in a low-energy effective field theory is critically discussed. Violation of these constraints implies superluminal propagation, in the sense that the low-frequency limit of the phase velocity v ph ( 0 ) exceeds c. It is explained why causality is related not to v ph ( 0 ) but to the high-frequency limit v ph ( ∞ ) and how these are related by the Kramers–Kronig dispersion relation, depending on the sign of the imaginary part of the refractive index Im n ( ω ) which is normally assumed positive. Superluminal propagation and its relation to UV completion is investigated in detail in three theories: QED in a background electromagnetic field, where the full dispersion relation for n ( ω ) is evaluated numerically and the role of the null energy condition T μ ν k μ k ν ⩾ 0 is highlighted; QED in a background gravitational field, where examples of superluminal low-frequency phase velocities arise in violation of the positivity constraints; and light propagation in coupled laser–atom Λ-systems exhibiting Raman gain lines with Im n ( ω ) < 0 . The possibility that a negative Im n ( ω ) must occur in quantum field theories involving gravity to avoid causality violation, and the implications for the relation of IR effective field theories to their UV completion, are carefully analysed.